Fifth International Workshop on the Seismic Performance of Non-Structural Elements (SPONSE)
Modelling One-dimensional Rolling Response of Rigid Bodies on Casters Using Physics Engine
Simulation
Cong Xu1, Quincy Ma2, and Masahiro Kurata3
1 PhD Candidate, Dept. of Civil & Environmental Engineering, The University of Auckland, NEW ZEALAND
2 Senior Lecturer, Dept. of Civil & Environmental Engineering, The University of Auckland, NEW ZEALAND
3 Associate Professor, Disaster Prevention Research Institute, Kyoto Univ., Kyoto 6110011, JAPAN [email protected]
Abstract. In hospitals, many contents and medical equipment such as incubators, ventilators and anaesthetic machines are placed on casters. During an earthquake, they may exhibit complex 3-dimensional motion, including rolling, sliding, swivelling, rocking and a combination of the above. There are limited studies on the dynamics of rigid objects on casters. Due to the complexity and sensitivity of this problem, it is difficult to have an accurate yet easy to use model for engineers to predict the seismic performance of building contents. This preliminary study explores the use of physics engine based simulation for the modelling the one-dimensional rolling response of such objects. A subset of 50 simple rolling experiments are reported herein, outlining the process from establishing the rolling resistance relationship to establishing the required parameters required for successful simulations in Unity. A custom programming script has been developed to model the rolling resistance force. A comparison of the Unity simulation results with the physical testing confirms the accuracy of the physics engine simulation and confirms the potential of the use of physics engine simulation for more complex scenarios.
Keywords: Physics engine simulation, Non-structural components, Casters & Wheels, Rolling equipment, Dynamic modelling.
1.Introduction
Hospitals serve an important post-disaster function and are expected to remain operational during and after a major earthquake. Past earthquakes have shown that the performance of non-structural elements (NSEs) significantly affects hospital operations, and recent experience is that NSEs in hospitals have not been performing satisfactorily in earthquakes (FEMA, 2015; Fierro et al., 2011; Gould & Marshall, 2012; Miranda et al., 2012; Motosaka & Mitsuji, 2012). Past surveys showed that up to 80% of the investment of a building lies in non-structural elements and equipment. In cases of hospitals, the portion is even higher and reaches 90% (Taghavi & Miranda, 2003). Because of the large financial investment, NSEs damage typically contributes significantly to buildings’ direct economic losses resulting from earthquakes (Miranda et al., 2012). Many NSEs are unanchored for moveability requirements in hospitals, making them prone to overturning during earthquakes. The overturning of partition walls and shelves poses a particularly hazardous condition for patients and other occupants and they can block egress routes. Moreover, damage to medical equipment due to earthquakes can cause significant economic loss. Another concern is that medical equipment on casters or wheels could displace significantly during earthquakes, leading to collision with nearby objects and potentially overturning. Even without apparent damage, the accelerations from collisions can cause some acceleration-sensitive medical equipment to lose functionality. All the above reduce the ability of a hospital to treat injuries and save lives in major earthquakes.
The dynamic response of a rigid object on casters is more complex than it may first appear. A caster is a mechanical assembly comprising one or more wheels, an axle, a fork supporting the axle and wheel(s), and possibly other accessories such as brakes or locking devices (ANSI ICWM, 2018). There can be many different design parameters for a caster amongst all options, including wheel size, arrangement, stem offset, caster angle, wheel material, brakes, swivel or rigid casters, plate or stem mounting and mounting positions.
Figure 1 (Left) illustrates the definition of some of these parameters.
Figure 1 Definitions of caster parameters (Left) and Schematic distribution of elementary forces along the contact surface of a hard rolling cylinder and a soft horizontal underlay (Right) (Vozdecký et al., 2014).
A caster can roll, skid, swivel or a combination of all modes simultaneously. Objects supported on casters can exhibit three-dimensional translation, rotation and rocking motion. All these lead to the complex dynamic motions of the system.
Although extensive studies have covered the dynamics of rocking objects, studies focused on the objects on casters permitted to rock are rare. Chatzis and Smyth (2013) proposed a theoretical model for examining the seismic response of a rigid object with wheels that can swivel and roll on a flexible support medium.
They used a concentrated springs model to simulate the deformability of the support medium and a rolling- friction formulation to model the vertical and horizontal reaction forces. Although this theoretical model provided useful insights, it was mathematically complex and not easy for engineers to implement. Other
experimental endeavours exist, such as that by Nikfar and Konstantinidis (2017, 2019), which subjected an ultrasound machine on two non-swivelling wheels and two casters and a cart with four casters to earthquake ground motion using a shaking table, and that by Hutchinson et al. (2013), with involved full-scale testing of a five story reinforced concrete building including equipment on casters.
Researchers have also applied classical finite element analysis (FEA) to model the caster rolling problem (Jose Luis Martin, 2015). However, owing to the complex interactions between rocking, rolling and sliding modes, the sensitivity of the problem to parameters selection, the large geometric nonlinearity and the problem’s history dependency, simulations often are very computation resource intensive, difficult to validate, and difficult to be transferred from one situation to another.
Physics engine simulation is a modelling technique that relies on Newtonian mechanics to simulate an object’s dynamic response (Boeing & Braunl, 2007; Laurell, 2008). It is a powerful and user-friendly tool that has been successfully applied to simulate the motion of the complex system, including robotics (Degrave et al., 2019), civil engineering (Izadi & Bezuijen, 2018), particle simulation (He & Zheng, 2020), medical training (Ricardez et al., 2018), and even disaster simulation (Kim et al., 2016). Physics engine simulation has been shown to be able to accurately model a rigid block’s complex rocking behaviour (Ma et al., 2018). However, limited research is available validating its use to simulate the dynamic response of equipment on casters.
This study advances the use of physics engine simulations for accurately modelling rigid bodies on casters.
The study focuses on modelling medical equipment on casters and other common objects found in hospitals. This paper presents sets of preliminary characterisation experiments involving the use of three casters types and two different floor coverings. The experiments focus on gaining insight on casters’ critical parameters, such as rolling resistance on several floor coverings. This paper sets the scene for future research considering more complex dynamics.
This study utilises Unity, a physics engine based software, to conduct dynamic computer simulations. Unity’s simulation results are compared to the experimental results for a rolling trolley to assess Unity’s capability.
50 constant force pulling tests are included in this study. A motion capture (MoCap) system measured the 3-dimensional displacement of the trolley during each test.
2.Background
2.1 PHYSICS ENGINE
Physics engines are computer software that simulates the dynamics of physical systems following a modern adaptation of Discrete Element Method. A key tenant of the underlying analysis is dividing a system into interacting bodies and particles. Physics engines are widely applied to game development, virtual reality systems, and the film industry to produce realistic Computer Generated Imagery (CGI). There is generally a trade-off between high precision and the speed of simulation. Physics engines have also been used in accurate scientific applications, such as examples found in computation fluid dynamics and geotechnical engineering (Götz et al., 2010; He et al., 2021). The substantial computational capacity offered by GPUs’
parallel processing architecture makes it possible to expedite the rigid body simulation of numerous rigid bodies, which was previously challenging to complete in real-time (Nguyen, 2008). Rolling problems are not well suited to structural finite element framework, due to the large displacement, low stiffness, and changing boundary conditions nature of the problem. Physics engine simulations in contrast are well suited for simulating this type of problems. Basic rolling behaviour can be modelled using Unity’s built-in functions.
Physic engines consist of four main subsystems: contact detection, contact resolution, force computation, and state integration (Hecker, 2000). A schematic of these subsystems is shown in Figure 2. Unity uses Nvidia PhysX as its physics engine for 3D projects.
Figure 2 The modules of the physics engines adapted from (Laurell, 2008).
2.2 MOTION CAPTURE SYSTEM
MoCap system uses markers attached to the objects to capture their real-time displacements. For this study, a marker-based motion capture system captured submillimetre-accurate 3D measurements of the caster trolley motion. The system consisted of six Optitrack Prime 41 cameras. It operated on the infrared spectrum and at a sampling rate of 120 frames per second. Using a MoCap system simplified instrumentation and ensured the instrumentation did not influence the specimen motion.
2.3 ROLLING PHYSICS THEORY
The difficulty of simulating the dynamic response of castor objects stems from the compounding challenge of sensitive and nonlinear components. On an individual component level, rolling wheels and twisting casters are low-stiffness and thus prone to numerical instability and initial condition sensitivity. When castors operate in an assembly, this activates additional global degrees of freedom whose responses are strongly history-dependent. For instance, inaccurate tracking of the twisting of a single castor will cause the object to change course and leads to erroneous subsequent motion prediction. Likewise, for incorrect tracking of wheel speed, wheel sliding and interaction with future forcing.
Vozdecký et al. (2014) present a thorough review of several mechanics-based analytical rolling models.
These models can be categorised into three groups based on whether the roller and/or the underlay medium deforms during the rolling motion. Some commonly used models include the hard body rolling along a deformable underlay that has symmetric deformation (Bilobran & Angelo, 2013) or a soft body rolling on a hard underlay (Cross, 2015).
Rolling resistance (fr) is defined in this paper as the generalised nonconservative force that resists the rolling motion as a wheel rolls along a surface. The exact cause of rolling resistance needs to be established. Past research has attributed it to sources such as the deformation and hysteresis of wheels and ground in motion (Wong, 2001), frictional resistance of joints, micro slip and friction on the contact surface and surface
adhesion (Ai et al., 2011). Many factors can affect rolling resistance. The most significant factors include wheel load, wheel diameter, wheel material, flooring material, and floor conditions (roughness, cleanliness, slope) (Lippert & Spektor, 2013). This study adopts Equation (1) as the empirical relationship between the rolling resistance force (fr), the coefficient of rolling resistance (Cr), the wheel radius (R) and the normal force (N) for a rolling wheel.
r r
f C N
= R (1) The underlying assumption for the study aligns with the idealised deterministic rolling model as summarised in Vozdecký (2014). This model assumes a rigid wheel is rolling on the deformable ground, with the ground providing an undetermined distribution of elementary forces. Vozdecký model assumes the sum of the elementary forces to a resultant elementary force F , which drives the wheel acceleration. The rolling resistance force (fr) in Equation (1) is the x component of the resultant elementary force (
F
r x, ) in Vozdecký model. A free-body diagram of the forces of the arrangement is shown in Figure 1 (Right).3.Pulling Test Procedure
A series of pulling tests were carefully conducted in the laboratory. A total of 50 pulling test trials were carried out. These tests cover possible caster types, caster arrangement, flooring and initial conditions combinations for the purpose of establishing simulation parameters for the rolling problem. The test setup was designed to ensure each trial was repeatable and minimise uncertainties in boundary conditions.
The test data aims to calibrate the corresponding settings in a Unity rolling trolley simulation model. The parameters of interest include the static friction coefficient 𝜇𝑠, kinetic friction coefficient 𝜇𝑘, coefficient of restitution e, and the twisting resistance force. These correspond to Static Friction, Dynamic Friction, Bounciness and hinge joints’ Spring and Damper values in Unity. There is no corresponding setting or mechanism to model rolling resistance in Unity. Thus, a C# script was created to translate the rolling resistance coefficient into a constantly updating force that is applied to each wheel.
Two vinyl floor coverings and three caster types are selected for testing. The floorings are commercial products currently used in new patient wards and corridors in New Zealand hospitals. The selected casters have wheel sizes and materials that are common in practice. All casters in this study can swivel. The detailed caster specifications are presented in Table 1.
Table 1 Casters specifications.
ID Wheel Material
Wheel Diameter Wheel Width Caster Height Stem Offset
R100 Institutional Rubber 100mm 32mm 131mm 84mm
R125 Institutional Rubber 125mm 32mm 160mm 100mm
P100 Polyurethane 100mm 32mm 131mm 84mm
3.1 EXPERIMENT SETUP
The rolling specimen for the experiment is a 96.5 kg steel trolley, with its centre of mass located at its planar geometric centre 487 mm above the bottom of the base plate. The trolly rest on the floor covering of interest, which is in turn secured to a level concrete floor. Concrete pavers with a mass of 13.6 kg each are
placed onto the trolley in some trials to simulate varying wheel normal force. Detailed dimensions of the trolley are shown in Figure 3.
During each test trial, the trolley is set into motion by releasing the trolley restraint abruptly. This allows a drop weight connected with the trolley via a steel cable and pulley arrangement to apply a constant force to the specimen. Figure 3 shows a schematic of the pulling test setup. A load cell is connected in series with the pre-tensioned steel cable, and it records the real-time pulling force acting on the trolley. Two string potentiometers record the displacements of the trolley and the drop weight. An accelerometer sits on top of the trolley and it measures the trolley acceleration during the pulling test.
Six OptiTrack cameras are placed around the specimen as the principal measurement system. One of the cameras is located down low and focuses on capturing the motion of the wheels. The system produces ± 0.10 mm accurate 3D position measurements at 120 Hz, of 12 selected locations of the setup designated by reflective markers.
Figure 3 Left: The pulling test setup. Right: Vinyl floor coverings used in the pulling test (Polyflor, 2014).
3.2 RESULTS
There are four phases of trolley motion in a typical test trial.
I. The trolley is held still and in static force equilibrium. The constant force of the drop weight is resisted by blocks placed in front of the casters.
II. The blocks are released and the trolley is set into motion. The trolley accelerates due to the unbalanced force provided by the drop weight via the cable and pulley arrangement. The drop weight falls freely to the ground applying a nearly constant force.
III. The drop weight reaches the ground, and the steel cable no longer applies any force to the trolley.
The trolley slowly decelerates due to rolling resistance and other frictional losses while experiencing the very small tension of the string potentiometer.
IV. The test is terminated when either the trolley reaches the end of the flooring or it comes to a rest.
Mystique Classic Forest FX
String Pot
String Pot
Load cell MC Camera
Trolley Rig
1100mm
550mm 1470mm
Pulley
Pulley
Steel cable
Figure 4 shows the trolley’s indicative displacement, velocity and acceleration time history and the cable force time history during a particular test trial.
The rolling resistance force provided by the casters can be evaluated by considering the dynamic forces equilibrium of the system. The condition of which is summarised in the free body diagram and the corresponding equations are shown in Figure 5. It is noteworthy that the string potentiometers’ force is accounted for and removed during preliminary data processing, so it is not included in the figure.
The time history response shows that the setup successfully achieved the constant force and constant acceleration condition during the pulling phase (Phase II). Furthermore, the rolling resistance can be calculated from the measured pulling forces and the acceleration data following the equations in Figure 5.
Average acceleration over Phases II and III instead of instantaneous values are used for the calculation to minimise the effects of measurement errors.
Figure 6 presents a plot of rolling resistance fr against the wheel’s normal force N. Both figures support the chosen theoretical rolling resistance relationship shown in Equation (1). In these experiments, the floor covering choice has no effect on fr . Figure 7 (Left) shows the average rolling resistance force plots in the test trials for two wheel diameters with the same wheel material and two different flooring materials. This confirms rolling resistance decreases with wheel diameters. Figure 7 (Right) plots the rolling resistance force against velocity for tests using the R100 casters. Within the test parameters, rolling speed had no effect on fr. Figure 8 shows the Unity user interface and the trolley model for this study. The trolley Unity model consists of an assembly of rigid bodies. Figure 9 shows a schematic of how they are connected and related to each other. Hinge joint connections enable the swivelling of the wheel fork and the rotating of each wheel about its axle.
Figure 4 Time history response of a pulling test with 100mm diameter rubber wheel caster.
Figure 5 Free body diagram of the trolley.
Figure 6 The total normal force on all wheels against rolling resistance.
Figure 7: Calculated fr for two wheel diameters (Left) and different trolley speed (Right).
Figure 8 Unity interface and the trolley model.
Figure 9 Trolley model hierarchy in Unity.
Figure 10 Different colliders used in the trolley model and the use of joints.
4.Physics Engine Model Simulation and Results
4.1 CHOICE OF COLLIDERS
Collisions between objects are key to physics engine simulations. Collision takes place when one GameObject makes contact with another GameObject. When a Collision event occurs, the function OnCollisionEnter is invoked (Unity, 2020). This enables users to develop custom programming to model bespoke behaviour.
Users define a collider for each object in Unity, and colliders define the shape and interfaces where collisions can occur(Unity, 2020). The selection of colliders affects the temporal and spatial accuracy of collision detection, and subsequently affects the overall simulation accuracy. There are five choices of standard 3D colliders in Unity: box collider, capsule collider, sphere collider, terrain collider and mesh collider. There is
no cylinder collider in Unity. This study has consequently adapted a capsule collider to approximate the rotating wheels. The rotational inertia of the wheel is manually modified using rigid bodies’ inertiaTensor property, from that calculated for a capsule to match the correct inertial value for a cylinder. This ensures correct energy transfer in the simulation.
Different parts of the trolley (each GameObject) use different colliders, as shown in Figure 10.
• The trolley’s body uses the box collider.
• The wheels use capsule colliders.
• The ground uses the box collider.
4.2 MODELLING ROLLING RESISTANCE
Unity does not have a specific parameter or setting for modelling rolling resistance. In this project, a C#
script was developed to provide a rolling resistance force fr to each wheel. The rolling resistance, as shown in Equation 1, has two key factors, i) a dimensionless parameter Cr/R as established by the trendline in Figure 6, and ii) the normal force on each wheel N. Unity’s built-in 3D physics engine calculates the normal force on each wheel by detecting collision forces between colliders on the wheel and the ground. Only the vertical component of the collision force is used in the fr calculation. An “OnCollisionStay” function is triggered whenever the wheel-collider and ground-collider collide and remain in contact for a timestep. This allows the rolling resistance force to be applied to the wheel in the opposite direction of the relative movement between the wheel and the ground.
All phases except for phase I of the pulling tests are simulated in Unity. In Phase II, the constant pulling force measured from the physical experiment is applied to the Unity model. This force is applied for the same duration as the physical experiment. Following this, in Phase III, a constant string potentiometer force is applied the model, consistent with the puling experiment.
4.3 SIMULATION AND EXPERIMENTAL RESULTS COMPARISON
Figure 11 shows a typical time history comparison between the Unity simulation and the physical experiment results. It shows that the application of hinged joints, capsule colliders and custom-scripted rolling resistance force in Unity can effectively simulate the real trolley motion. The Unity simulation also enables mapping of the energy transfer between the different components during a pulling test, as shown in Figure 11. The WF line represents the work done by the drop weight and the string potentiometer. The WR line represents the work done by the simulated resistance force, and EKT is the kinetic energy of the trolley.
Figure 11 Top: Typical comparison of Unity simulation and physical experiment time histories (Left: displacement, Right: velocity), Bottom: Energy content during a pulling test.
5.Discussion and Conclusion
This study experimentally established the rolling resistance coefficient of a caster trolley on two vinyl floor coverings. A key finding was confirming rolling resistance force is proportional to the wheel’s normal force and inversely proportional to wheel diameter. During this experiment, the rolling resistance coefficient did not vary with wheel speed. This study showed advanced energy dissipation behaviour can be simulated through Unity flexible programming functions. Unity accurately simulated the physical pulling tests with the appropriate modifications to its basic settings and functions. The unique arrangement of capsule colliders with hinge joints modelled the caster’s behaviour well.
It is worth noting that although Unity was used to only model a simple pulling test in this study, it has the potential to simulate more complex dynamic responses of caster objects subjected to various ground motions. Simulations and experiments with swivelling castor motion and multi-directional rolling motion are not reported in this paper due to length limitations.
6.Acknowledgement
The authors extend their thanks to the undergraduate students, Caleb Knight-Polamalu, Daniel van Dam, Louie Love-Parata, Megan Hoare and William Wang, for their assistance during the experiments. Sincere thanks also to the China Scholarship Council for providing the PhD scholarship for the first author.
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